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CENTER OF MASS AND MOMENT OF INERTIA     15£
It is evident from this theorem that when the lamina is rotated about the 2-axis Ix and Iy change, in general, but their sum remains constant.
122.   Theorem II. — The moment of inertia of a body about any axis equals its moment of inertia about a parallel axis through the center of mass
plus the product of the mass of the body by the square of the distance between the two axes.
Let the axis be perpendicular to the plane of the paper   and  pass   through the point 0, Fig. 86.   Further let dm be any ele-                     FlG 86 ment of mass, r its distance from the axis through 0, and rc its distance from i parallel axis through the center of mass, C.    Then if < denotes the distance between the axes we have
/= /  r2dm Jo
rm
= /   (Tc2 + a2—2 arc cos6) dm Jo
/*m                           /»TM.                                  /*?n
= /   r2 dm + I   a2 dm — 2 a I   rc cos 6 dm
Jo               Jo                   Jo
/»m
= Ic + ma2 — 2 a I   x dm.
Jo
x dm = mx
and in the present case the center of mass is at the origin therefore x and consequently the last integral vanishes Thus we get
I = Ic+ma2.                             (IV
123.   Radius of Gyration.—The radius of gyration of a bod] with respect to an axis is defined as the distance from th<